Lemma 79.9.7 (06E3)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 79: More on Groupoids in Spaces
Section 79.9: Properties of groups over fields and groupoids on fields
Lemma 79.9.7 (cite)

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Lemma 79.9.7. In Situation 79.9.2. If $r : \mathop{\mathrm{Spec}}(k) \to R$ is a morphism such that $s \circ r, t \circ r$ are automorphisms of $\mathop{\mathrm{Spec}}(k)$ , then the map

$R \longrightarrow R, \quad x \longmapsto c(r, x)$

is an automorphism $R \to R$ which maps $e$ to $r$ .

Proof. Proof is identical to the proof of More on Groupoids, Lemma 40.10.6. $\square$

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